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Random response evaluation of mono-stable and bi-stable Duffing systems with Dahl friction

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Abstract

The random responses of Duffing systems with dynamic friction are investigated, and a uniform procedure is established to derive the explicit expressions of stationary probability densities for mono-stable potential and bi-stable potential cases. The Dahl friction model is adopted to describe the dynamic friction behavior, and the friction force is first expressed as a time-domain integration of the velocity function by directly integrating the auxiliary differential equation with non-smooth term. By introducing the associated generalized harmonic transformations for mono-stable potential and bi-stable potential cases, the friction force is further simplified and only approximately depends on the present velocity, amplitude and phase. The friction force is then mandatorily separated as conservative and dissipative components through the quasi-linearization technique. At last, the mono-stable and bi-stable Duffing systems with dynamic friction are equivalent to approximate nonlinear systems with modified damping and modified stiffness. The stationary probability densities of the original friction systems are evaluated by those of the approximate nonlinear systems, which can be analytically expressed. The effectiveness and accuracy of the uniform procedure are verified by comparison with the Monte Carlo simulation for mono-stable potential (including hardening stiffness and softening stiffness) and bi-stable potential cases.

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Acknowledgments

Y.W. acknowledges the National Natural Science Foundation of China under Grant Nos. 11472240 and 11302064, and the Fundamental Research Funds for the Central Universities under Grant No. 2016FZA4025. X.L. acknowledges the National Natural Science Foundation of China under Grant No. 11202181. Z.L. acknowledges the National Natural Science Foundation of China under Grant Nos. 1153000141 and 11321202.

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Authors and Affiliations

  1. The State Key Laboratory of Fluid Power Transmission and Control, Department of Engineering Mechanics, Zhejiang University, Hangzhou, China

    Y. Wang, X. L. Luan, X. L. Jin & Z. L. Huang

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  1. Y. Wang
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  2. X. L. Luan
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  3. X. L. Jin
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  4. Z. L. Huang
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Correspondence to X. L. Jin.

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Wang, Y., Luan, X.L., Jin, X.L. et al. Random response evaluation of mono-stable and bi-stable Duffing systems with Dahl friction. Arch Appl Mech 86, 1827–1840 (2016). https://doi.org/10.1007/s00419-016-1147-3

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  • DOI: https://doi.org/10.1007/s00419-016-1147-3

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